Journal of Integer Sequences, Vol. 6 (2003), Article 03.3.4

How to Differentiate a Number

Victor Ufnarovski
Centre for Mathematical Sciences
Lund Institute of Technology
P.O. Box 118
SE-221 00 Lund

Bo Åhlander
Electrum 213
164 40 Kista

We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibnitz rule. The aim of the article is to consider the basic properties of this map and to show how to generalize the notion to the case of rational and arbitrary real numbers. We make some conjectures and find some connections with Goldbach's Conjecture and the Twin Prime Conjecture. Finally, we solve the easiest associated differential equations and calculate the generating function.

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(Concerned with sequence A003415 .)

Received April 4, 2003; revised version received July 27, 2003. Published in Journal of Integer Sequences September 17, 2003.

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